Optimal. Leaf size=20 \[ \frac {1}{3} \log (x) \left (x^2-\log \left (\frac {1}{3}+4 x\right )\right ) \]
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Rubi [A]
time = 0.26, antiderivative size = 30, normalized size of antiderivative = 1.50, number of steps
used = 11, number of rules used = 8, integrand size = 54, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {1607, 6820,
12, 2404, 2341, 2354, 2438, 2439} \begin {gather*} \frac {1}{3} x^2 \log (x)+\frac {1}{3} \log (3) \log (x)-\frac {1}{3} \log (x) \log (12 x+1) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 1607
Rule 2341
Rule 2354
Rule 2404
Rule 2438
Rule 2439
Rule 6820
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x^2+12 x^3+\left (-12 x+2 x^2+24 x^3\right ) \log (x)+(-1-12 x) \log \left (\frac {1}{3} (1+12 x)\right )}{x (3+36 x)} \, dx\\ &=\int \frac {1}{3} \left (x+\frac {2 \left (-6+x+12 x^2\right ) \log (x)}{1+12 x}-\frac {\log \left (\frac {1}{3}+4 x\right )}{x}\right ) \, dx\\ &=\frac {1}{3} \int \left (x+\frac {2 \left (-6+x+12 x^2\right ) \log (x)}{1+12 x}-\frac {\log \left (\frac {1}{3}+4 x\right )}{x}\right ) \, dx\\ &=\frac {x^2}{6}-\frac {1}{3} \int \frac {\log \left (\frac {1}{3}+4 x\right )}{x} \, dx+\frac {2}{3} \int \frac {\left (-6+x+12 x^2\right ) \log (x)}{1+12 x} \, dx\\ &=\frac {x^2}{6}+\frac {1}{3} \log (3) \log (x)-\frac {1}{3} \int \frac {\log (1+12 x)}{x} \, dx+\frac {2}{3} \int \left (x \log (x)-\frac {6 \log (x)}{1+12 x}\right ) \, dx\\ &=\frac {x^2}{6}+\frac {1}{3} \log (3) \log (x)+\frac {\text {Li}_2(-12 x)}{3}+\frac {2}{3} \int x \log (x) \, dx-4 \int \frac {\log (x)}{1+12 x} \, dx\\ &=\frac {1}{3} x^2 \log (x)+\frac {1}{3} \log (3) \log (x)-\frac {1}{3} \log (x) \log (1+12 x)+\frac {\text {Li}_2(-12 x)}{3}+\frac {1}{3} \int \frac {\log (1+12 x)}{x} \, dx\\ &=\frac {1}{3} x^2 \log (x)+\frac {1}{3} \log (3) \log (x)-\frac {1}{3} \log (x) \log (1+12 x)\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.02, size = 26, normalized size = 1.30 \begin {gather*} \frac {1}{3} \left (x^2 \log (x)+\log (3) \log (x)-\log (x) \log (1+12 x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.58, size = 25, normalized size = 1.25
method | result | size |
risch | \(-\frac {\ln \left (x \right ) \ln \left (4 x +\frac {1}{3}\right )}{3}+\frac {x^{2} \ln \left (x \right )}{3}\) | \(19\) |
default | \(\frac {\ln \left (3\right ) \ln \left (x \right )}{3}+\frac {x^{2} \ln \left (x \right )}{3}-\frac {\ln \left (x \right ) \ln \left (12 x +1\right )}{3}\) | \(25\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 21, normalized size = 1.05 \begin {gather*} \frac {1}{3} \, {\left (x^{2} + \log \left (3\right )\right )} \log \left (x\right ) - \frac {1}{3} \, \log \left (12 \, x + 1\right ) \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.45, size = 18, normalized size = 0.90 \begin {gather*} \frac {1}{3} \, x^{2} \log \left (x\right ) - \frac {1}{3} \, \log \left (4 \, x + \frac {1}{3}\right ) \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.13, size = 20, normalized size = 1.00 \begin {gather*} \frac {x^{2} \log {\left (x \right )}}{3} - \frac {\log {\left (x \right )} \log {\left (4 x + \frac {1}{3} \right )}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 24, normalized size = 1.20 \begin {gather*} \frac {1}{3} \, x^{2} \log \left (x\right ) + \frac {1}{3} \, \log \left (3\right ) \log \left (x\right ) - \frac {1}{3} \, \log \left (12 \, x + 1\right ) \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.36, size = 16, normalized size = 0.80 \begin {gather*} -\frac {\ln \left (x\right )\,\left (\ln \left (4\,x+\frac {1}{3}\right )-x^2\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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